Complexity Theory. Jörg Kreiker. Summer term Chair for Theoretical Computer Science Prof. Esparza TU München

Transcription

1 Complexity Theory Jörg Kreiker Chair for Theoretical Computer Science Prof. Esparza TU München Summer term 2010

2 Lecture 4 NP-completeness

3 Recap: relations between classes EXP PSPACE = NPSPACE conp NP conp P NL NP L

4 Agenda efficiently checkable certificates reductions, hardness, completeness Cook-Levin: 3SAT is NP-complete

5 Certificates NP: efficiently checkable certificates NP computable with NDTM in polynomial time.

6 Certificates NP: efficiently checkable certificates NP computable with NDTM in polynomial time. Theorem (Certificates) For every L {0, 1} holds: L NP if and only if there exists a polynomial p : N N and a polynomial-time TM M such that for every x {0, 1} x L u {0, 1} p( x ). M(x, u) = 1

7 Certificates NP: efficiently checkable certificates NP computable with NDTM in polynomial time. Theorem (Certificates) For every L {0, 1} holds: L NP if and only if there exists a polynomial p : N N and a polynomial-time TM M such that for every x {0, 1} x L u {0, 1} p( x ). M(x, u) = 1 M is called verifier u is called certificate

8 Certificates NP: efficiently checkable certificates NP computable with NDTM in polynomial time. Theorem (Certificates) For every L {0, 1} holds: L NP if and only if there exists a polynomial p : N N and a polynomial-time TM M such that for every x {0, 1} x L u {0, 1} p( x ). M(x, u) = 1 M is called verifier u is called certificate Proof: certificate is sequence of choices NDTM guesses certificate

9 Certificates Examples Indset: certificate is set of nodes, size of certificate for k nodes in graph with n nodes O(k log n) 0/1 ILP: given a list of m linear inequalities with rational coefficients over variables x 1,..., x k ; find out if there is an assignment of 0s and 1s to x i satisfying all inequalities; certificate is assignment. Iso: given two n n adjacency matrices; do they define isomorphic graphs; certificate is a permutation π : [n] [n]

10 Certificates Agenda efficiently checkable certificates reductions, hardness, completeness Cook-Levin: 3SAT is NP-complete

11 Completeness Reductions reminder IF there is an efficient procedure for problem A and and an efficient procedure for B using the procedure for A THEN B cannot be radically harder than A notation: B A We have seen (at least) two reductions. 3 Coloring was reduced to Indset the diagonalized, undecidable language reduced to Halt reduction does not make anything smaller

12 Completeness Reductions definition Definition (Karp reduction) Let L, L {0, 1} be languages. L is polynomial-time Karp reducible to L iff there exists a polynomial-time computable funtion f : {0, 1} {0, 1} such that for all x {0, 1} We write L p L. x L f(x) L

13 Completeness Reductions definition Definition (Karp reduction) Let L, L {0, 1} be languages. L is polynomial-time Karp reducible to L iff there exists a polynomial-time computable funtion f : {0, 1} {0, 1} such that for all x {0, 1} We write L p L. x L f(x) L Note: p is a transitive relation on languages (because the composition of polynomials is a polynomial).

14 Completeness Hardness and Completeness Definition (NP-hardness and -completness) Let L {0, 1} be a language. L is NP-hard if L p L for every L NP L is NP-complete if L is NP-hard and L NP.

15 Completeness Hardness and Completeness Definition (NP-hardness and -completness) Let L {0, 1} be a language. L is NP-hard if L p L for every L NP L is NP-complete if L is NP-hard and L NP. Examples of NP-hard languages: Indset, Halt k, Halt

16 Completeness Hardness and Completeness Definition (NP-hardness and -completness) Let L {0, 1} be a language. L is NP-hard if L p L for every L NP L is NP-complete if L is NP-hard and L NP. Examples of NP-hard languages: Indset, Halt k, Halt Observation L NP-hard and L P implies P = NP L NP-complete implies L P iff P = NP

17 Cook-Levin Do NP-complete languages exist? upcoming result independently discovered by Cook (1971) and Levin (1973) uses notion of satisfiable Boolean formulas Boolean formula ϕ over variables X = {x 1,..., x k } defined by ϕ ::= x ϕ ϕ ϕ ϕ ϕ write x instead of x, x and x literals u assume formulas are in CNF: ϕ = disjunctions j u ij called clauses formula is in k-cnf if the no clause has more than k literals i j u ij

18 Cook-Levin Cook-Levin Theorem ϕ is satisfiable iff there exists an assignments a : X {0, 1} making ϕ true 3SAT = {ϕ ϕ in 3-CNF and satisfiable} Theorem 3SAT is NP-complete.

19 Cook-Levin Proof agenda 1. SAT is NP-complete (without restriction to clauses of size three) 1.1 SAT, 3SAT NP 1.2 for every L NP L p SAT 2. Show that SAT p 3SAT

20 Summary What have we learnt? NP is polynomial certificates Karp reductions, hardness, completeness Cook-Levin: reduce any language in NP to 3SAT up next: more NP-complete problems, P vs. NP, tool demos